We also will not explore the relation between the Hubbard model and experiments in any particular quantum material. Nonetheless, much of the modern resurgence of interest in this model is a consequence of the role it has played in the study of high temperature superconductivity in the cuprates. Originally, this connection was suggested on rather basic phenomenological grounds [1]}, [2]} - the “parent state” in the cuprates is a quasi-2d antiferromagnetic insulator, and the Hubbard model on the square lattice is the simplest possible model of a doped antiferromagnet. It is thus striking that many of the properties of this model that have been uncovered in subsequent theoretical studies turn out to resemble essential features of the electronic states of the cuprates that have been revealed by experiments, viz.: 1) The Hubbard model (at least for a range of \(U\)  not too large) exhibits a “dome” of \(d\) -wave superconductivity throughout a range of doping. 2) It exhibits antiferromagnetic long-range order when undoped. 3) It evolves to an effectively weakly-coupled Fermi liquid state upon heavy doping. 4) It exhibits a tendency toward formation of long-period unidirectional charge and spin density wave orders (stripes) in the same range of doping in which superconducting correlations (at least locally) appear strongest. Indeed, as in the materials themselves, the relation between density-wave and superconducting orders appears more complicated than the collocation “competing orders” might suggest, leading to their description as “intertwined.” As has been noted earlier [3]}, this congruence in behavior is sufficiently striking that it encourages the view that the solution of the Hubbard model can, in some sense, be considered to be the solution of the high \(T^{\vphantom{\dagger }}_{\rm c}\)  problem.
